Track / Stage 5 / 248 of 400 #848 of 1964

Problem 848

AIME late

Geometry Difficulty 5.7 Prove it

468. Prove that a line passing through the midpoint of side $AB$ of triangle $ABC$ and its incenter divides the segment connecting vertex $C$ with the point of tangency of the inscribed circle with side $AB$ in half.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

468. Four points $A, C_{1}, B$ and $C$ are vertices of a degenerate circumscribed quadrilateral; therefore, the midpoint $M_{3}$ of side $A B$ and the midpoint $N$ of segment $C C_{1}$ are midpoints of the diagonals and lie on the same line with the center of the inscribed circle (see theorem 21).

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.